2.2 If \( y=\sqrt[6]{100000} \), WITHOUT USING A CALCULATOR, determine the value \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \) in terms of \( y \).

To solve the problem, we start by determining \( y = \sqrt[6]{100000} \). First, express \( 100000 \) as a power of 10: \[ 100000 = 10^5 \] Then, \( y \) becomes: \[ y = \sqrt[6]{10^5} = 10^{5/6} \] Next, we calculate \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \). 1. **Calculate \( \sqrt[3]{16} \)**: \[ 16 = 2^4 \quad \Rightarrow \quad \sqrt[3]{16} = \sqrt[3]{2^4} = 2^{4/3} \] 2. **Calculate \( \sqrt[3]{625} \)**: \[ 625 = 5^4 \quad \Rightarrow \quad \sqrt[3]{625} = \sqrt[3]{5^4} = 5^{4/3} \] 3. **Calculate \( \sqrt{10} \)**: \[ \sqrt{10} = 10^{1/2} \] Now, we can combine these results: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 2^{4/3} \times 5^{4/3} \times 10^{1/2} \] Since \( 10 = 2 \times 5 \), we can write: \[ 10^{1/2} = (2 \times 5)^{1/2} = 2^{1/2} \times 5^{1/2} \] Thus, \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 2^{4/3} \times 5^{4/3} \times 2^{1/2} \times 5^{1/2} \] Now, we combine the powers of 2 and 5 separately: \[ = 2^{4/3 + 1/2} \times 5^{4/3 + 1/2} \] To combine the exponents, convert them to a common denominator (which is 6): \[ 4/3 = 8/6 \quad \text{and} \quad 1/2 = 3/6 \] Thus, \[ 4/3 + 1/2 = 8/6 + 3/6 = 11/6 \] For \( 5 \): \[ 4/3 + 1/2 = 8/6 + 3/6 = 11/6 \] So we have: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 2^{11/6} \times 5^{11/6} = (2 \times 5)^{11/6} = 10^{11/6} \] Now we relate this back to \( y \): \[ y = 10^{5/6} \] To express \( 10^{11/6} \) in terms of \( y \): \[ 10^{11/6} = 10^{5/6} \times 10^{6/6} = y \cdot 10 = 10y \] Finally, we arrive at the answer: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 10y \]